

\chapter{Hidden differences}
\small
\begin{quote}
Bose's theory of radiation and my analogous theory of ideal
gases have been reproved by Mr. Ehrenfest and other colleagues
because in these theories the quanta or molecules are not
treated as structure statistically independent of one another,
without this circumstance being especially pointed out in our
papers.  This is entirely correct.  If one treats the quanta as
being statistically independent of one another in their
localization, then one obtains the Wien radiation law; if one
treats the gas molecules analogously, then one obtains the
classical equation of state for ideal gases, even if one
otherwise proceeds exactly as Bose and I have\cite{Einstein1925}.\\
--Albert Einstein
\end{quote}
\normalsize
Quantum statistics, a subject which has played a central
role in the development of quantum mechanics, is only rarely mentioned in
quantum information.  It is usually not necessary to know whether a qubit is a
boson or a fermion in order to understand a quantum circuit.  On the contrary,
one of the central achievements of quantum information science has
been to abstract away the particular physical systems in which quantum
circuits are implemented and so to select model systems where the fermionic or bosonic
character of the particles involved plays no role.  The circuit
model of quantum computing wherein each qubit sits on a separate rail
and interacts with other qubits via gates leaves no room for the exchange effects of
quantum statistics.

While this abstraction has been useful in teasing out the
information-theoretic aspects of the quantum theory, when it comes to
real-world implementations of quantum ideas, the particles that
carry quantum information have a quantum statistical nature that
often cannot be ignored.  The quantum-statistical nature of
real-world particles is important in many systems ranging from
superconductors to neutron stars, so it should come as no surprise
that it also arises in linear optics quantum
information\cite{KLM2001}.  In linear optics quantum computing the
`interactions' between photons are regulated by
the quantum statistical Hong-Ou-Mandel effect.  While post-selection
is used to select those experimental trials where only one photon
occupies each output mode of the beamsplitter, those occasions
where the post-selection `fails' result in multiple photons occupying
a single spatio-temporal mode.  This means that quantum statistics
lies just below the surface of most post-selected linear quantum optics
experiments where photons carry the qubits, including the demonstration of quantum teleportation\cite{Bouwmeester1997}, the
non-deterministic CNOT\cite{OBrien2004}, the generation of GHZ states\cite{Pan2000} and
the one-way quantum computer\cite{Walt2005}. 

For this reason, it is important that techniques developed for quantum
state estimation and quantum state tomography be made compatible with the
quantum-statistical nature of states.  This, as it turns out, is a
non-trivial task, and one to which the remainder of this chapter will
be devoted.  Along the way we will establish an operational
distinction between fundamental indistinguishability and
indistinguishability caused merely by an inability to measure
differences.  We will introduce an accessible density matrix for an
experimental system that contains all the information about a system
that can be ascertained given a limited number accessible of degrees of freedom
for a particle.  Using this tool, we will be able to determine how
distinguishing information in the state will affect all future
measurements.  Finally, we will put this theory into practice by
applying it to two and three-photon systems.

\section{Two-mode systems}
Before we can examine the effects of distinguishability and quantum
statistics in systems of few photons, we need an adequate
description of photons as indistinguishable particles.  The quantum
theory of optics treats each optical mode
(i.e. each linearly independent solution to Maxwell's equations) as an
independent linear harmonic oscillator.  The quantum excitations of these
oscillators are called photons.  The orthogonal polarization modes
$\ket{H}$ and $\ket{V}$ represent one important set of modes; other modes
include, for instance, the Hermite-Gauss modes of Gaussian cavities
and the different arms of a Mach-Zehnder interferometer.  

We consider a system with a fixed number of photons $N$.  These photons
can be distributed between two modes in any way so that the quantum
state will be a superposition of states of the form $\ket{m,N-m}$
where $m$ is the number of photons in the first mode and $N-m$ is the
number of photons in the second mode.

It was first pointed out by Schwinger\cite{Sakurai} that two uncoupled simple harmonic
oscillator modes such as two optical polarization modes, are formally
equivalent to a spin system.  If one of these modes is the horizontal
polarization mode and the other the vertical mode, we can define the
usual raising and lowering operators with the commutation
relations $\left[a_H,a_H^\dagger\right]=1$ and
$\left[a_V, a_V^\dagger \right]=1$, but with
$\left[a_H,a_V^\dagger\right]=0$ since the two modes do not interact.
We can also define operators that couple the modes together $J_+=\hbar
a_H^{\dagger} a_V$ and $J_-=\hbar a_V^\dagger a_H$.  Rather than
commuting like $a$ and $a^\dagger$, these two operators satisfy
$\left[J_+,J_-\right]=2\hbar J_z$ where $J_z=\frac{\hbar}{2}\left(a_H^\dagger a_H
-a_V^\dagger a_V\right)$.  It can also be shown that
$\left[J_z,J_{\pm}\right]=\pm \hbar J_{\pm}$.  This set of commutation
relations is identical to the ones for angular momentum, and so the
entire quantum mechanical machinery devoted to analyzing angular
momentum can also be applied to these states.

For the specific case of photon polarization, the angular momentum
operators have historically been replaced with Stokes
operators\cite{Shumovsky1998} which give the total number of photons
($S_0$) and the degree of polarization along the $D-A$ ($S_1$), $R-L$
($S_2$) and $H-V$ ($S_3$) axes of the Poincar\'e sphere 
\begin{align}
S_0&=a_H^\dagger a_H+a_V^\dagger a_V\\
S_1&=\frac{1}{2}\left(a_H^\dagger a_V+a_V^\dagger a_H\right)\\
S_2&=\frac{1}{2i}\left(a_H^\dagger a_V-a_V^\dagger a_H\right)\\
S_3&=a_H^\dagger a_H-a_V^\dagger a_V.
\end{align}

\subsection{Characterizing two-mode states of light}
We now have a formalism capable of describing multi-photon
polarization states as angular momentum states.  It seems that it
should be easy to adapt the methods of quantum state tomography to
this description and to thereby measure the density matrix on the
angular momentum Hilbert space.  A problem arises, though, when one
considers what measurements are needed to perform the tomography.
Take, for example, the simple case of a system of two photons.  Via
the Schwinger formalism, the polarization can easily be seen to
form a spin-$1$ system.  If we were to do a projective measurement
onto the diagonal elements of this system we would project the state
onto $a^\dagger_H a^\dagger_H$, $a^\dagger_H a^\dagger_V=a^\dagger_V a^\dagger_H$ and
$a^\dagger_V a^\dagger_V$ or, rewriting these second-quantized states
as first-quantized Dirac kets through the use of Clebsch-Gordan
coefficients, 
\begin{align}
\ket{j=1,m=1}&=\ket{HH}\\
\ket{j=1,m=0}&=\frac{1}{\sqrt{2}}\left(\ket{HV}+\ket{VH}\right)\\
\ket{j=1,m=-1}&=\ket{VV}.
\end{align}
The problem lies in the $\ket{j=1,m=0}$ term which is a maximally-entangled state of the two
single-photon polarizations.  It is a little strange to think of
single-photon polarizations when the photons are fundamentally
indistinguishable, but in this case we can think of a simple
operational meaning for single-photon polarization.  We can
deterministically split the $H$ and the $V$ photons at a polarizing
beamsplitter into two different spatial modes $1$ and $2$, and then
the particles in these two modes can be safely thought of as qubits
and the polarization correlations between them will be exactly those predicted for
the maximally-entangled Bell state $\ket{\psi^+}$.

If we reverse this reasoning then it seems that simply measuring one
of the photons to be $H$ and the other to be $V$ should be enough to
project onto the state $\ket{j=1,m=1}$, but this seems too good
to be true.
Typically projections onto maximally-entangled states
are difficult and involve multi-photon interference and
post-selection.  Clearly no such interference or post-selection is
needed if all we are doing is counting $H$ and $V$ photons.  Can such
a measurement really be viewed as a projection onto a
maximally-entangled state?

Let us assume that our measurement detects only polarization, but is
completely insensitive to the other properties that may
define different modes for the two photons.  Now imagine that
one of the photons is blue
and the other is red.  Then, rather than having a single
state $a^\dagger_H a^\dagger_V$, one has two different states
$a^\dagger_{H,b} a^\dagger_{V,r}$ and $a^\dagger_{H,r}
a^\dagger_{V,b}$.  Surprisingly, this difference in the colour of the two
photons can lead to different results in the polarization
measurements.  

Say, for example that we measure the rate at which we will detect both
photons to be diagonally polarized.  We can rewrite $a_H^\dagger
a_V^\dagger$ as 
\begin{align}
a_H^\dagger a_V^\dagger \ket{vac} &=\left[\frac{1}{\sqrt{2}} \left(a_D^\dagger+a_A^\dagger
\right)\right]\left[\frac{1}{\sqrt{2}} \left(a_D^\dagger-a_A^\dagger
  \right)\right]\ket{vac}\notag \\
&=\frac{1}{2} \left(a_D^\dagger a_D^\dagger+a_D^\dagger
a_A^\dagger-a_A^\dagger a_D^\dagger-a_A^\dagger a_A^\dagger
\right)\ket{vac}\notag \\
&=\frac{1}{\sqrt{2}} \left(\ket{2_D,0_A}-\ket{0_D,2_A}
\right).
\end{align}
The probability of the state containing two diagonal photons can
immediately be seen to be $50\%$.  On the other hand if the two
photons are different colours so that the state is $a^\dagger_{H,b}
a^\dagger_{V,r}$ then we have 
\begin{align}
a_{H,b}^\dagger
a_{V,r}^\dagger \ket{vac} &=\left[\frac{1}{\sqrt{2}} \left(a_{D,b}^\dagger+a_{A,b}^\dagger
\right)\right]\left[\frac{1}{\sqrt{2}} \left(a_{D,r}^\dagger-a_{A,r}^\dagger
  \right)\right]\ket{vac}\notag \\
&=\frac{1}{2} \left(a_{D,b}^\dagger a_{D,r}^\dagger+a_{D,b}^\dagger
a_{A,r}^\dagger-a_{A,b}^\dagger a_{D,r}^\dagger-a_{A,b}^\dagger a_{A,b}^\dagger
\right)\ket{vac}\notag \\
&=\frac{1}{\sqrt{2}}
\left(\right.\ket{1_{D,b},1_{D,r},0_{A,b},0_{A,r}}+\ket{1_{D,b},0_{D,r},0_{A,b},1_{A,r}}\\
&-\ket{0_{D,b},1_{D,r},1_{A,b},0_{A,r}}-\ket{0_{D,b},0_{D,r},1_{A,b},1_{A,r}}\left.\right).
\end{align}
In this case the probability of obtaining two diagonal photons is
$25\%$.  Even though our measurement apparatus is insensitive to
colour, the mere existence of distinguishing colour information 
can affect the outcome of polarization measurements.

It should be clear from this example that counting $H$ and $V$ photons
does not project the state onto $\ket{j=1,m=0}$.  Instead it projects
the state onto the subspace of states having one $H$ and one $V$
photon, or equivalently onto the space of states with $m=0$.  Although
for \emph{indistinguishable} photons the only polarization state with
$m=0$ is the $\ket{j=1,m=0}$ state, if the photons are distinguishable
the state $\ket{j=0,m=0}$ is also allowed.  Insofar as
the goal of tomography is to provide the best estimate of the quantum
state, it cannot be assumed from the outset that the photons are
indistinguishable.  Rather tomography should measure all the information
available about the state and try to determine from that whether the
photons are indistinguishable or not.

If this seems a trivial insight, consider that the first quantum state
tomography scheme\cite{Bogdanov2004_1,Bogdanov2004_2} applied to states of two `indistinguishable' photons
only characterized the $j=1$ subspace and assumed that the measurement
of an $H$ and a $V$ photon constituted a projection onto the state
$\ket{j=1,m=0}$.  A careful examination of the data in
\cite{Bogdanov2004_2} shows rather
poor agreement between the density matrices and the measurements
which may possibly be due to this incorrect assumption.

This makes state estimation for indistinguishable particles
fundamentally different from state estimation for distinguishable
particles.   With distinguishable particles one can safely ignore those degrees
of freedom that don't encode information.  For example when qubits are
encoded in polarization one can safely ignore wavelength and spatial
mode degrees of freedom of the photon and measure the polarization
density matrix.  If the polarization state is correlated to these
other degrees of freedom, the result is simply a reduction in the
purity of the state.  

For indistinguishable particles that may have hidden distinguishing
information, the unmeasured degrees of freedom can not only affect
purity, but can even change the effective size of the Hilbert space!
As we have seen, for two particles, instead of only having to consider
the $j=1$ space, distinguishing information can expand the state into
the $j=0$ space as well.  Luckily, with a few group-theoretic
tricks, the effect of distinguishing information can be understood
within the context of the density matrix description of the quantum
state.  Through the next several sections we will develop the group
theoretical tools needed to understand distinguishable and
indistinguishable quantum particles and use them to develop a new
framework for quantum state tomography that can address cases of
particles with hidden distinguishability.  

\subsection{Symmetric measurements}
Let us assume that we are given one end of an optical fiber.  At the
other end is a black-box source emitting pulses containing $N$
photons.  We have equipment capable of separating the polarization
modes and counting how many photons are in each mode.  We also have
waveplates that allow us to perform arbitrary unitary rotations of
the polarization modes prior to measurement.  What can we learn about
the state of the photons from our measurements?  Or, slightly
rephrased, what range of density matrix descriptions of the state will
be consistent with all the possible measurements taken with this
system?

To answer this question we first need to consider what range of states
are allowed by the laws of physics.  The most important restriction on
the allowed states is given by the bosonic nature of photons which
implies that if all of the properties of two of the photons in the
state are exchanged then the state vector must be unchanged.
Determining which states satisfy this requirement will
necessitate a departure into group theory, the relevant aspects
of which will be developed in the next few sections.  

\subsection{Group theory}

\subsubsection{The symmetric group}
The possible permutations of $N$ particles form a mathematical group
called the symmetric group denoted by $S_N$.  As a reminder, the
statement that the permutations form a group means that they are closed under
composition, that all elements have an inverse, that
there is an identity element and that composition is associative.
This can easily be seen by considering the possible permutations of
three numbers, $\left(1,2,3\right)$.  Permuting the first two numbers
gives $\left(2,1,3\right)$, the second two $\left(1,3,2\right)$, the
first with the second, the second with the third and the third with
the first $\left(3,1,2\right)$ and so on.  In fact the permutations
can be written down in just this way, by labeling how each permutation
in $S_N$ affects the order of $N$ particles.  For example, the six
permutations in $S_3$ can be written as:
\begin{align}
&\left(
\begin{array}{ccc}
1 & 2 & 3\\
1 & 2 & 3\\
\end{array}
\right),\\
&\left(
\begin{array}{ccc}
1 & 2 & 3\\
2 & 1 & 3\\
\end{array}
\right),\left(
\begin{array}{ccc}
1 & 2 & 3\\
1 & 3 & 2\\
\end{array}
\right),\left(
\begin{array}{ccc}
1 & 2 & 3\\
3 & 2 & 1\\
\end{array}
\right),\\
&\left(
\begin{array}{ccc}
1 & 2 & 3\\
3 & 1 & 2\\
\end{array}
\right),\left(
\begin{array}{ccc}
1 & 2 & 3\\
2 & 3 & 1\\
\end{array}
\right).
\end{align}      
One may obtain a more compact expression of these permutations by
resolving them into `cycles'.  A cycle $(r_1,r_2,\dots,r_n)$ is a
permutation that replaces each element $r_k$ by the element $r_{k+1}$
that follows it except for the last element $r_n$ which is replaced by
$r_1$.  The six permutations in $S_3$ listed above can be resolved
into cycles as
\begin{align}
\left\{(1)(2)(3),(12)(3),(1)(23),(13)(2),(123),(321)\right\}.
\end{align}
We say that two permutations containing the same number of cycles and
whose cycles are of the same length belong to the same class.  This is
to say that they can be transformed into each other by a `symmetry
transformation' that maps each permutation element to another
permutation element of the same class.

There are three permutation classes in $S_3$, one containing
$(1)(2)(3)$, one containing $(12)(3),(1)(23),(13)(2)$ and one
containing $(123),(321)$.
\subsubsection{Representations}
A representation of a group is a mapping between the elements of the
group and a set of matrices that have the same relationships under
multiplication as the group elements do under composition.  It can be
shown that any group can be represented by a set of unitary matrices
and in particular, the symmetric group $S_N$ can always be represented
by a set of symmetric matrices (which gives the group its name).  To
give an example for $S_3$, we can
make the following correspondences between permutations and matrices:
\begin{align}
\label{S3irrepexample}
(1)(2)(3)\rightarrow \left(\begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right) \\
(12)(3)\rightarrow \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)\\
(1)(23)\rightarrow \left(\begin{array}{cc} -\frac{1}{2} & \frac{1}{2}\sqrt{3} \\ \frac{1}{2}\sqrt{3} & \frac{1}{2}\end{array}\right)\\
(13)(2)\rightarrow \left(\begin{array}{cc} -\frac{1}{2} & -\frac{1}{2}\sqrt{3} \\ -\frac{1}{2}\sqrt{3} & \frac{1}{2}\end{array}\right)\\
(123)\rightarrow \left(\begin{array}{cc} -\frac{1}{2} & \frac{1}{2}\sqrt{3} \\ -\frac{1}{2}\sqrt{3} & -\frac{1}{2}\end{array}\right)\\
(321)\rightarrow \left(\begin{array}{cc} -\frac{1}{2} & -\frac{1}{2}\sqrt{3} \\ \frac{1}{2}\sqrt{3} & -\frac{1}{2}\end{array}\right).
\end{align}

The reader can check that the matrices have the same multiplicative
relations as the permutations to which they correspond.

The representation of $S_3$ given above is by no means unique.  One
could easily construct another set of matrices having the same
multiplicative relations by, for example, taking each matrix ${\bf A}$
and constructing a three-by-three matrix 
\begin{align}
\label{reducible_representation}
\left(\begin{array}{cc} \bf{A} &
  0\\ 0 &1\end{array}\right).
\end{align}

Clearly such a transformation adds extra dimensions to the matrices
without changing the multiplicative relationships between the
matrices. 
Such representations are called
\emph{reducible}, while representations that cannot be put into the
form of $\ref{reducible_representation}$ through a unitary
transformation on the representation elements are called 
\emph{irreducible representations} or \emph{irreps} for short.     

Even more simply, one could assign each permutation to the
number $1$.  Since $1\times 1=1$, all the multiplicative relations
between the permutations are trivially satisfied.  This is called the
trivial irrep.  While this
representation is \emph{irreducible} it is called \emph{unfaithful}
whereas a representation like table \ref{S3irrepexample} that assigns a
unique matrix to each group element is called \emph{faithful}.

It can be shown that the number of distinct irreducible
representations of a group is equal to the number of classes in the
group, that is to say the sets of group elements that can be exchanged
without changing the multiplicative relationships in the group.
Again, for the symmetric group the number of classes is simply the
number of different possible lengths and numbers of cycles.

The dimensionality of an irrep refers to the
dimensionality of its matrices.  Thus the irrep of table
\ref{S3irrepexample} has dimension two while the trivial irrep has
dimension one.

For any symmetric group $S_N$ there will always be two one-dimensional
irreps, the trivial representation and another irrep called the
alternating irrep that can be obtained by assigning to each group
element either the number 1 or -1 consistent with the multiplicative relationships.  The name alternating comes from
the fact that one can properly assign 1 to all elements obtained by an
even number of permutations of two particles and -1 to those obtained
through an odd number of permutations.  All the other irreps of $S_N$ are multidimensional.  

We mention these two irreps because they play a special role in
quantum statistics.  It is a fact of nature that fundamental particles
are identical.  That is to say that no physical measurement can tell
apart two multiparticle quantum states in which the two of the
particles are exchanged along with all the properties of those
particles (spatial, temporal and polarization properties for photons,
say).  In group theory language, all physical observables of the system must be
invariant under the action of any permutation operator.  This means
that the quantum states of the system must be invariant up to a global
phase.  At most, a permutation of all the degrees of freedom of two
particles will multiply a state by a unit modulus constant.  Thus the
states must transform according to one of the two one-dimensional
irreducible representations of the symmetric group.  We call particles
transforming under the trivial representation bosons and particles
transforming under the alternating representation fermions.  When the quantum
state of a system transforms according to a particular irrep of a
group we say that it \emph{carries} that irrep.

This is not to say that these two representations are the only useful
ones in quantum mechanics.  Often we are concerned with the
permutation of only some degrees of freedom of particles in which case
fundamental particles no longer need to be regarded as identical
because the remaining degrees of freedom can still distinguish them.
A classic example of this is the generalized Hong-Ou-Mandel effect in which the
spatial modes of some number of photons are exchanged at a beamsplitter that has
no effect on the polarization properties of the photons.  The
polarization state at the output of such a beamsplitter is not limited
to the trivial and alternating group but can carry other irreducible
representations as well. 


\subsubsection{Counting the irreps}
The total number of distinct irreps of $S_N$ can be obtained by simple
counting arguments.  The number of irreps corresponds to the number of
classes which is to say the number of different ways that the numbers
from $1$ to $N$ can be divided into
cycles.  This is called the number of partitions of $N$ i.e. the
number of different ways that we can select a set of integers
$\left\{\lambda_i \right\}_k$ such that
$\lambda_1+\lambda_2+\dots+\lambda_k=N$.

A convenient way of labeling the partitions is to use Young
diagrams, a series of boxes arranged according to the rule that the
number of boxes in each row must be equal to or less than the number
in the previous row.  Let's consider the three partitions of $3$
listed below:
\begin{align}
\begin{array}{cc}
1+1+1 & \yng(3)\\
2+1 & \yng(2,1)\\
3 & \yng(1,1,1)
\end{array}
\end{align}.
To each of these Young diagrams there corresponds a distinct partition
of the numbers from $1$ to $N$ and hence a distinct irreducible
representation of $S_N$.  

Young diagrams can be used to obtain the dimensionality of each of the
irreps of $S_N$ through the construction of \emph{Young tableaux}.  A
standard Young tableau is the arrangement of the number from $1$ to $N$ in a
Young diagram so that the numbers in any row or column of the table
are non-decreasing.  

We state without proof the following theorem\\
\emph{
The number of dimensions of an irrep of $S_N$ represented by a
particular Young diagram is equal to the number of distinct standard
Young tableaux that it can accommodate }
Figure \ref{standard_young_tableaux_example} shows the five standard
Young tableaux for the Young diagram $\yng(3,3)$
\begin{align}
\label{standard_young_tableaux_example}
\young(123,456),\young(124,356),\young(135,246),\young(125,346),\young(134,256)
\end{align}
In the case of Young diagrams with two rows the number of distinct
standard Young tableaux can be shown from simple combinatorics to be $\left(\begin{array}{c}
  n\\k \end{array} \right)-\left(\begin{array}{c}
    n\\k-1 \end{array}\right)$\footnote{$\left(\begin{array}{c}n
        \\k\end{array}\right)$ is the number of combinations of $k$
        elements selected from among $n$ elements. $\left(\begin{array}{c}n
        \\k\end{array}\right)=\frac{n!}{k!(n-k)!}$} where $n$ is the
    total number of blocks in the tableau and $k$ is the number of blocks in the
    second row.  For the example given in figure
    (\ref{standard_young_tableaux_example}) we have
\begin{align} 
\left(
\begin{array}{c}
6\\
3 
\end{array}
\right)
-\left(
\begin{array}{c} 
6\\
3-1 
\end{array} 
\right)&=
  \frac{6!}{3!3!}-\frac{6!}{2!4!} \\
&=20-15\\
&=5.
\end{align}
The irreps corresponding to two-row Young diagrams are especially
important in considering the permutations of two-level systems such as
photon polarizations.  

While one can also use Young tableaux to help construct a linearly
independent basis spanning the irreps of $S_N$ it is easier to make
use of a very powerful duality between the irreps of $S_N$ and those
of $SU(2)^{\otimes N}$ described in the next section\footnote{In fact the duality holds between any GL(n), but
  we will only make use of it for SU(2).}.

\subsubsection{The Schur-Weyl duality}
Now that we have developed the theory of permutations as applied to
quantum states we are ready to use it to describe a system of identical
photons.  We will consider a system of photons each of which carries
with it a $m$-dimensional Hilbert space that completely describes its
properties (polarization, spatial mode and frequency mode).  The
limitation to a finite-dimensional Hilbert space may
seem strange since properties like spatial mode and frequency are
often considered as continuous variables, but since our interest is
in tomography we will have to discretize the Hilbert space at some
point in order to have a hope of
measuring the state of the system.  Moreover, our main concern in the
experiments that will be presented is with polarization which, as a
two level system, is already finite-dimensional.

The Hilbert space of a system of $N$ photons will be the tensor
product of the $N$ spaces describing the individual photons
$\mathcal{H}_m^{\otimes N}$.  To make clear the action of permutations
we would like to find a set of basis vectors for this space that carry
the irreps of $S_N$.  Luckily the Schur-Weyl duality\cite{Hammermesh} in group theory
shows that this is always possible.  Their theorem can be stated as 
\begin{align}
\mathcal{H}_m^{\otimes N}
 =  \bigoplus_{\lambda} \mathcal{Q}^m_\lambda \otimes \mathcal{P}_\lambda .
\end{align}
This decomposes the Hilbert space $\mathcal{H}_m^{\otimes n}$ into a
direct sum of tensor products between spaces
$\mathcal{Q}^m_\lambda$ that carry an irreducible representation of
$U(m)$ and $\mathcal{P}_\lambda$ that carries a particular irreducible
representation of $S_N$.  The theorem states that this decomposition
is multiplicity-free so that each irrep of $U(m)$ and $S_N$ appears
exactly once in the sum and so can be labeled with the same index
$\lambda$.  

In the case where $m=2$ we have a particularly instructive example,
both because it nicely describes photon polarization and
because the irreps of $U(2)=U(1)\otimes SU(2)$ are easy to work with because of their
relationship to the irreps of the group of spatial rotations generated
by angular momentum operators\footnote{While we already derived this
  from the Schwinger formalism, a group theorist would say that this is
  because SU(2) is the universal covering group of O(3), the group of spatial
  rotations in 3-dimensional space.  The two groups share a common Lie
algebra and any irrep of O(3) is also an irrep of SU(2).}  

This being the case, we can label the irreducible representations of
SU(2) with a \emph{total angular momentum} $j$ which is either an integer or a half-integer.  The irrep labeled $j$ will
be $2j+1$ dimensional.  This is the same $j$ that we discussed in the
context of the Schwinger formalism.  Its basis states states are labeled with
an index $m$ that ranges from $-j$ to $j$.  The action of SU(2) on
these states will generally map a state $\ket{j,m}$ onto a linear
combination of states $\sum_{m=-j}^j c_m \ket{j,m}$ all with the same
$j$.  

The $N$ copies of SU(2) present in the Hilbert space
$\mathcal{H}_2^{\otimes N}$ of $N$ photon
polarizations allow irreps with $j$ between $0$ and $N/2$ when $N$ is
even and between $1/2$ and $N/2$ when $N$ is odd.  The Schur-Weyl
theorem in this case is  
\begin{align}\label{SchurWeyl}
\mathcal{H}_2^{\otimes N}
 =  \bigoplus_{j=0\textrm{ or } 1/2}^{N/2} \mathcal{\mathcal{Q}}_j \otimes \mathcal{P}_j.
\end{align}
A set of basis states taking advantage of this decomposition can be
created using the machinery of Clebsch-Gordan for building states
carrying a specific irrep $j$ out of tensor products states.  

One familiar example of this is the singlet-triplet
decomposition of the space of two two-level states,
$\mathcal{H}_2\otimes
\mathcal{H}_2=\mathcal{H}_{j=1}\oplus\mathcal{H}_{j=0}$.  The
Schur-Weyl theorem express the fact that this decomposition, constructed to
yield states with well defined total angular momentum $j$, also carry
irreps of $S_2$. 

Using polarization as our two level system so that $m=1/2$ is labeled
with $H$ and $m=-1/2$ with $V$ we can write out this decomposition
explicitly as
\begin{align}
&\ket{j=1,m=1}=\ket{HH}\\
&\ket{j=1,m=0}=\frac{1}{\sqrt{2}}\left(\ket{HV}+\ket{VH}\right)\\
&\ket{j=1,m=-1}=\ket{VV}\\
&\ket{j=0,m=0}=\frac{1}{\sqrt{2}}\left(\ket{HV}-\ket{VH}\right).
\end{align}
Notice that the $j=1$ states all carry the trivial representation of
$S_2$, $\yng(2)$ since if the two particle labels are permuted the
states are unchanged.  The $j=0$ state carries the other irrep of
$S_2$, the antisymmetric irrep $\yng(1,1)$, and will be multiplied by
one if the identity permutation is applied and by -1 if the polarizations
are permuted. 

If polarization is the \emph{only} degree
of freedom describing the photons then the requirement that the whole
state be invariant under permutations would only allow the $j=1$
states.  The $j=0$ state does not have the requisite symmetry to be a
valid boson state (although it would be a valid state for fermions
whereas the $j=1$ states would not).  In the second-quantized
formalism this fact is built into the raising and lowering operator
commutation (and anti-commutation) relations.  Usually this is thought
of as a convenient property of the formalism, but when
non-trivial permutation symmetries are possible due to hidden distinguishability
it can lead to confusion\footnote{Two different referees of our 
  paper on this topic objected that all of our results would be trivial if we
  would only have used second-quantized notation!} and to incorrect
results such as the Bogdanov tomography scheme's equating of a
measurement of $m$ with a projection onto $\ket{j=N/2,m}$.
\ignore{
A more complicated example is obtained when we have four particles.
We can take four copies of $\mathcal{H}_2$,
$\mathcal{H}_2\otimes\mathcal{H}_2\otimes\mathcal{H}_2\otimes\mathcal{H}_2$
and construct the $2^4=16$ states that span the space while carrying irreps
of $SU(2)$ and $S_N$  
}
%Insert 4-particle decomposition here

With the Schur-Weyl theorem establishing that we can span the
Hilbert space of $N$ particles with states that carry irreps of both
$U(m)$ and $S_N$ we now wish to take the decomposition one step
further.  Among the $m$ levels describing the complete state of each
particle we would like to separate the $d$ levels that are visible
to our experimental measurements from the $D$ levels that are hidden
from them.  Clearly $D \times d=m$, and for the moment we will also require
that $D \geq d$ which will make the math easier and, in any event is
certainly true in the experiments we will be considering\footnote{In
  the case where $d \geq D$ we can always formally add more hidden
  levels until $d \geq D$ without affecting the derivation.}.  In this
case we can apply the Schur-Weyl theorem separately to the visible and
hidden degrees of freedom. 

\begin{align}\label{HVSW}
\mathcal{H}_m^{\otimes N}&=\left(\mathcal{H}_{\mathrm{vis}}^{\otimes
    d}\right)\otimes \left(\mathcal{H}_\mathrm{hid}^{\otimes D}\right)\\
&= \left( \bigoplus_{\lambda} \mathcal{Q}^d_\lambda \otimes \mathcal{P}_\lambda \right)\otimes
   \left( \bigoplus_{\lambda'} \mathcal{Q}^D_{\lambda'} \otimes \mathcal{P}_{\lambda'} \right)\,,
\end{align}
A general state will be a superposition of states in these spaces, so
that 
\begin{align}
\ket{\psi}=\sum_{\lambda} c_\lambda \ket{\lambda, q^d_\lambda, p_\lambda}_{\textrm{vis}} \ket{\lambda, q^D_\lambda,
p_\lambda}_{\textrm{hid}}.
\end{align} 

If we are dealing with bosons, however, this is \emph{too} general
because we have not yet applied the restriction that bosonic states
must be invariant under the action of all permutations operators,
which is to say that they must carry the trivial irrep of $S_N$   

One obvious way of achieving this is to restrict the kets to those
that carry the trivial irrep of $S_N$ in both the visible and hidden
degrees of freedom.  For those states a permutation will transform the
visible part trivially and the hidden part trivially, so clearly the
whole state will also transform trivially.

Can states which are invariant under all permutations also be constructed from
states that carry non-trivial irreps of $S_N$?  Consider a state that
carries the antisymmetric irrep of $S_N$ in both the visible and
hidden degrees of freedom.  Permutation operators on the visible and
hidden parts of the state act either by
leaving the state unchanged or by multiplying the state by -1.  If the
same permutation operator is applied to both the hidden and visible
degrees of freedom then either the whole state is multiplied by 1 or
by $(-1)(-1)=1$ so such a state is also invariant under all permutations.

What about multi-dimensional irreps of $S_N$?  Let's say we have
a state that carries a
mixed-symmetry irrep of $S_N$ like $\yng(2,1)$ for three photons.  The
$S_N$ irrep is two-dimensional and can be represented by
the matrices in table \ref{S3irrepexample}.  

We label the two states that transform according to these
measurements $\ket{a}_{\textrm{vis}}$ and $\ket{b}_{\textrm{vis}}$.
If the visible degree of freedom under consideration were polarization
then these states could be
\begin{align}
\ket{a}_{\textrm{vis}}&=\frac{1}{\sqrt{6}}\ket{HHV}+\frac{1}{\sqrt{6}}\ket{HVH}-\sqrt{\frac{2}{3}}\ket{VHH}\\
\ket{b}_{\textrm{vis}}&=\frac{1}{\sqrt{2}}\left(\ket{HHV}-\ket{HVH}\right).
\end{align}
The reader can verify that applying the permutations of $S_3$ to the
ordering of the polarizations does
indeed affect these states in the same way as the matrix
representation in table \ref{S3irrepexample}.

As long as the dimensionality of the space of hidden degrees of
freedom is not less than that of the visible degree of freedom ($D\geq d$) there
will also be states of the hidden degree of freedom carrying the
$\yng(2,1)$ irrep of $S_N$.  We may therefore create two states
$\ket{a}_{\textrm{hid}}$ and $\ket{b}_{\textrm{hid}}$ that
transform in the same way as $\ket{a}_\text{vis}$ and $\ket{b}_\text{vis}$ under the
action of the matrices in table \ref{S3irrepexample}.

Now if we construct the following state
\begin{align}
\frac{1}{\sqrt{2}}\left(\ket{a}_\textrm{vis}\ket{a}_\textrm{hid}+\ket{b}_\textrm{vis}\ket{b}_\textrm{hid}\right).
\end{align}
then it will be invariant under the action of all the permutations in
table \ref{S3irrepexample} (as long as the same permutation is performed
on the visible and hidden parts of the state).  That this is so
follows from the fact that all of the matrices of
table \ref{S3irrepexample} are orthogonal matrices.

Consider a particular matrix
\begin{align}
\mathbf M=\left(\begin{array}{cc}
q & r\\
s & t
\end{array}\right),
\end{align}
that takes $\ket{a}$ to $q\ket{a}+s\ket{b}$ and
$\ket{b}$ to $r\ket{a}+t\ket{b}$.  If the same matrix acts on both
parts of the state $\ket{a}\ket{a}+\ket{b}\ket{b}$ it will map it to  
\begin{align}
q^2\ket{a}\ket{a}+q s\ket{a}\ket{b}+s
q\ket{a}\ket{b}+s^2\ket{a}\ket{b}\\
+ r^2\ket{a}\ket{a}+r t\ket{a}\ket{b}+t
r\ket{a}\ket{b}+t^2\ket{a}\ket{b}\\
=(q^2+r^2)\ket{a}\ket{a}+(q s+r t)\ket{a}\ket{b}+(s
q+t r)\ket{a}\ket{b}+(s^2+t^2)\ket{a}\ket{b}.
\end{align}
For the mapping to leave the state unchanged we require that
\begin{align}
q^2+r^2&=1\\
q s +r t&=0\\
s^2+t^2&=1.
\end{align}
However, this is precisely the requirement that the columns of
$\mathbf M$ be orthonormal to each other.  Since all the
matrices in table \ref{S3irrepexample} are orthogonal, the state
$\frac{1}{\sqrt{2}}\left(\ket{a}_\textrm{vis}\ket{a}_\textrm{hid}+\ket{b}_\textrm{vis}\ket{b}_\textrm{hid}\right)$
is invariant under the action of all permutations in the $\yng(2,1)$ irrep.  

More generally, whenever we construct an equally-weighted
superposition of
kets that transform in the same way for the visible and hidden parts
over an irrep of $S_N$, we will always obtain a state that is invariant
under all permutations.  This follows directly from the fact that $S_N$ can
always be represented using orthogonal matrices and is a
straightforward generalization of the two-dimensional case
demonstrated above.  It also follows that such an equally
weighted sum is the \emph{only} linear
combination of basis vectors of a given irrep of $S_N$ that will
be invariant under all the permutations.

Using the notation introduced previously, any state of the form
\begin{align}
\frac{1}{\sqrt{\mathrm{dim}\mathcal{P}_\lambda}} \sum_{p_\lambda}
\ket{\lambda, q^d_\lambda, p_\lambda} \ket{\lambda, q^D_\lambda,
p_\lambda}
\end{align}
will be invariant under all permutations and hence will constitute a
valid state for bosons.

Finally, it should be clear that any state carrying a different irrep
in the visible and hidden degrees of freedom cannot be invariant under
all permutations since the two parts of the state will transform in
completely different way under permutations. 

The most general state of $N$ bosons can therefore be written as
\begin{align}
\label{generalstate}
\ket{\Psi^N_{d,D}} = \sum_\lambda \sum_{q^d_\lambda,q^D_\lambda}
A^{\lambda}_{q^d_\lambda,q^D_\lambda} \sum_{p_\lambda}
\ket{\lambda, q^d_\lambda, p_\lambda} \ket{\lambda, q^D_\lambda,
p_\lambda}\,,
\end{align}
where the $\lambda$ labels the terms in the Schur-Weyl
decomposition for the visible and hidden states respectively,
$q^d_\lambda$ runs over all the states in the $\lambda^{\text{th}}$
irrep of U(d) in the visible degree of freedom and $Q^d_\lambda$ does
the same for the $\lambda^\textrm{th}$ irrep of U(D) for the hidden
degree of freedom.  Since the only way to construct bosonic terms is
from states carrying the same irrep of $S_N$ in both the hidden and
visible degree of freedom, we can use the same index $\lambda$ to
label them both (rather than a separate indices $\lambda$ and $\lambda'$
for the visible and hidden parts). The final sum runs over the basis states for the
$\lambda^{\textrm{th}}$ $S_N$ irrep.  Note that the visible and hidden
part of the terms of this sum are represented by the \emph{same} basis
vector of the $S_N$ irrep.

The complex amplitudes $A^{\lambda}_{q^d_\lambda,q^D_\lambda}$ are
assumed to carry the normalization so as to satisfy the requirement
that
\begin{align}
\sum_\lambda
\mathrm{dim}\mathcal{P}_\lambda \sum_{q^d_\lambda,q^D_\lambda}
\vert A^{\lambda}_{q^d_\lambda , q^D_\lambda} \vert^2 = 1.
\end{align}

We'll take polarization as the visible
degree of freedom and make use of the total angular momentum to label
the SU(2) irreps.  We can therefore replace the label $\lambda$ with
$j$.  We'll assume that we have three photons so that $j$ runs from
$1/2$ to $3/2$.  We can also replace the index $\kappa$ with $m$ running
from $-j$ to $j$.  Finally we can label the basis states of $S_3$ with
an index label $s$.  The basis states for the hidden degrees of
freedom will, from the Schur-Weyl duality, also carry an irrep of
$S_3$, and there can be arbitrarily many orthogonal states carrying a
given $S_3$ irrep.  We can denote these many orthogonal states with
the labels $\lambda_j$ and $q_\lambda^D$.  Using these conventions, a properly bosonic state can be written as
\begin{align}
\sum_{j=1/2}^{3/2}\sum_{\lambda_j} \sum_{m=-j}^j \sum_{q^D_\lambda} c_{m,j;\lambda_j,q^D_\lambda}  \sum_{p_\lambda}
\ket{j,m,s} \ket{\lambda_j, q^D_\lambda,s}.
\end{align}
Note that the sum only consists of tensor products of states carrying
the same $S_3$ irrep $s$ in both the visible and hidden degrees of freedom.

\section{Hidden information}
In the last section we showed how to write down a state that is
invariant under all permutations and hence is an acceptable
description for a system of bosons.  In this section we will show what
information is accessible in the visible degrees of freedom once the
hidden degrees of freedom have been traced out.

In an experiment that measures the visible degree of freedom only, the
measurements we do can tell us about the state of the hidden
degrees of freedom only insofar as they correlate to the visible ones.
We project the state onto every possible state of the hidden degrees
of freedom and sum these projections to obtain a density matrix in the
visible degrees of freedom only which we will call the \emph{accessible
density matrix} $\rho^N_\text{d,acc}$.  
\begin{align}
\label{tracing_out_the_hidden_part}
\rho^N_{d,\text{acc}}
&= \text{Tr}_\text{hid}\left[ \ket{\Psi^N_{d,D}}\bra{\Psi^N_{d,D}}\right] \\
&= \sum_\lambda \sum_{q^d_\lambda,q'^d_\lambda} B^\lambda_{q^d_\lambda,q'^d_\lambda} \sum_{p_\lambda} \ket{\lambda,q^d_\lambda,p_\lambda}\bra{\lambda,q'^d_\lambda,p_\lambda}\,,
\end{align}
 where $B^\lambda_{q^d_\lambda,q'^d_\lambda}=\sum_{q^D_\lambda}
A^\lambda_{q^d_\lambda,q^D_\lambda}A^{\lambda\ast}_{q'^d_\lambda,q^D_\lambda}$.

It should be clear that since we started from the most general
possible boson state, the set of numbers $B^\lambda_{q^d_\lambda,q'^d_\lambda}$ provides an
informationally-complete description of the state of the visible
degrees of freedom.  Any property of the visible degree of freedom,
and the outcome of any measurement on them has to be a function of the
$B^\lambda_{q^d_\lambda,q'^d_\lambda}$ alone.

How many numbers $B^\lambda_{q^d_\lambda,q'^d_\lambda}$ are required to
describe the state?  It is clear from the structure of the formula
(\ref{tracing_out_the_hidden_part}) that each irrep of $S_N$ will
contribute one unique $B^\lambda_{q^d_\lambda,q'^d_\lambda}$ for each
unique pair $q^d_\lambda,q'^d_\lambda$.  Thus if the
$\lambda^\textrm{th}$ irrep of SU(d) is $k$-dimensional it will
contribute $k^2$ different numbers
$B^\lambda_{q^d_\lambda,q'^d_\lambda}$ regardless of dimensionality of
the $\lambda^\textrm{th}$ irrep of $S_N$.

We can make this even more precise by applying the Weyl character formula for
SU(N)\cite{Hammermesh}.  This formula gives the dimension of the $\lambda^{\text{th}}$ irrep of SU(N) as
% This I need to learn more about.  For the moment I'll just copy in
% Pete's expression.
\begin{align}
\mathrm{dim}(\lambda) = \prod_{1\leq i < j\leq d} \frac{\lambda_i -
  \lambda_j + j - i}{j - i},
\label{sumoverlambda}
\end{align}
where $\lambda$ is a partition of $N$ i.e. an ordered set of numbers $\lambda_n,
\lambda_{n-1},\ldots,\lambda_m$ such that $\lambda_i$ and $\sum_i
\lambda_i=N$ and $\lambda_i \geq \lambda_{i-1}$.  Because of the
one-to-one correspondence between partitions
and Young diagrams, this sum can be thought of as running over the
different Young diagrams with $N$ boxes.  

The SU(N) irrep associated with the trivial irrep of $S_N$ corresponds to the
partition of $N$ with only one non-zero $\lambda_i=N$ (i.e. a
partition $\left(N,0,0,\ldots,0\right)$.  The dimension of this irrep
can be written down explicitly as
\begin{align}
\prod_{j=2}^d \frac{N+j-1}{j-1} = {N+d-1 \choose N}.
\end{align}
When the particles in a bosonic state are fundamentally
indistinguishable, this is the full dimension of the Hilbert space
since there are no other degrees of freedom that can distinguish the
particles.  In all other cases where there are distinguishing, but
hidden, degrees of freedom, expression \ref{sumoverlambda} must be
summed over all $\lambda$ to arrive at total number of
$B^\lambda_{q^d_\lambda,q'^d_\lambda}$ in $\rho_{\text{acc}}$.  Thus
the number of free parameters in $\rho_{\text{acc}}$ is 
\begin{align}
\sum_{\lambda_i,\lambda_j} \prod_{1\leq i < j\leq d} \frac{\lambda_i - \lambda_j + j - i}{j - i}
= {N+d^2-1 \choose N},
\end{align}
as can be proved using the Cauchy formula for the general linear
group\footnote{This was shown by my co-author Peter Turner based on
  discussions with Trevor Welsh}.

For the case of polarization where $d=2$, this expression simplifies
to 
\begin{align}
\sum_{j=0 \, \mathrm{or} \, \frac{1}{2}}^{\frac{N}{2}} (2j+1)^2 = {N+3 \choose 3}.
\label{qubitdimensions},
\end{align}
so that, for example, three polarizations are completely described by
$20$ real numbers.

\subsection{Structure of the accessible density matrix}
Expression \ref{tracing_out_the_hidden_part} implies a particular
structure for the accessible density matrix.  If we consider for the
moment only SU(2), the structure of the density matrix will be
block-diagonal in angular-momentum space, with one or more $2j+1$ by $2j+1$
blocks for each allowed value of $j$.  For a three-photon polarization state
it would look like this:

\begin{equation}
\rho_{\text{acc}}=\left[ 
\begin{array}{ccc}
\left(
\begin{array}{cccc}
\ast & \ast & \ast & \ast \\
\ast & \ast & \ast & \ast \\
\ast & \ast & \ast & \ast \\
\ast & \ast & \ast & \ast \\
\end{array}\right)_{j=3/2} & & \\
&  \left(
\begin{array}{cc}
\ast & \ast \\
\ast & \ast \\
\end{array}\right)_{j=1/2}  & \\
&  & \left(
\begin{array}{cc}
\ast & \ast \\
\ast & \ast \\
\end{array}\right)_{j=1/2} \\
\end{array}
\right].
\label{formofrhoacc}
\end{equation}

The trace in equation \ref{tracing_out_the_hidden_part} eliminates any
coherences between sectors with different values of $j$.  This isn't
to say that such information does not exist, only that it will
be undetectable by measurements done on polarization and, by the same
token, will not affect any polarization operations one might like to
perform.  The reason for the elimination of these coherences is that
states with different values of $j$ necessarily carry different
representations of the symmetric group by the Schur-Weyl duality.
Saying that we are ignorant of the particle ordering (which is what we
do when we take the trace in equation
\ref{tracing_out_the_hidden_part}) applies all possible permutations to the
particles.  Coherences between states carrying two different irreps of
$S_N$ will necessarily transform differently under these permutation
and, it can be shown, will average to zero under all the
permutations.  

The trace also has an interesting effect on orthogonal vectors of
multi-dimensional irreps of $S_N$, for instance the spanning vectors
of the two $j=1/2$ spaces for three polarizations.  For these states
the trace in
equation \ref{tracing_out_the_hidden_part} reduces the equal
superposition over the orthogonal permutation spaces into an equal
convex sum over those spaces.  This means that two orthogonal states
with the same $j$ but carrying different permutation irreps will be
indistinguishable to polarization measurements.  The 
polarization measurements can only access the average value of the density matrix
in these two orthogonal spaces.  For three photons, the $16$ elements
of the $j=3/2$ space and the four
elements consisting of averages over the $j=1/2$ spaces exhaust the
$20$ free parameters calculated
from equation \ref{qubitdimensions}.   
  
A final interesting property of this accessible density matrix is that
the number of elements in it scales polynomially in the dimension of
the Hilbert space.  This means that particles that are distinguishable
in principle, but not in practice, form an intermediate case between
indistinguishable particles which are required to occupy states that
transform trivially under permutations and for which the Hilbert space
grows linearly in the number of particles and distinguishable
particles or qubits for which the Hilbert space grows exponentially.
The possibility of having particles distinguishable by hidden
information allows the state to contain more information than would be
possible if there were no distinguishability, but does not give
rise to the exponential scaling of Hilbert space size that is
considered essential to
quantum computation and other quantum information processing tasks.

\subsection{Implications for distinguishability}
In most experiments on photons there are two degrees of freedom of
importance.  One degree of freedom is the one in which the quantum
state is described and measurements are performed, and the other is a
distinguishing degree of freedom used to label the two photons.  The
prototypical quantum optics experiment is the Bell's inequality
violation\cite{Aspect1981} where two photons are identified by their different
propagation directions and measurements are performed on their
polarization.  What is really being measured here is a joint
correlation function between spatial mode and polarization.
This is entirely consistent with the classical statistical view that the photons
at the two locations are really `different' particles and that we are
measuring the polarization `of each photon'.

This all changes when photons travel in a single spatio-temporal mode, usually
defined by a single-mode optical fiber.  In some cases another degree
of freedom like wavelength can still be used as a labeling degree of
freedom\cite{Moreva2006}.  In most cases, though, the
goal is to get the photons into the same single mode and any
distinguishing information between the photons is there by
accident\cite{Mitchell2004,Eisenberg2004}. It is this sort of
experiment that this work addresses.  In such experiments one would
like a way of answering the question ``Is there, in principle, some
measurement on the degrees of freedom that I can't measure whose
outcome would distinguish one particle from the
other?''.  The discussion of the previous section helps both to make this
question operationally precise and to provide an answer.  

We first note that the visible degrees of freedom can at most provide
us with the $B^\lambda_{q^d_\lambda,q'^d_\lambda}$ experimental
density matrix elements.  At best, therefore, we can answer the
question, ``Are the density
matrix elements $B^\lambda_{q^d_\lambda,q'^d_\lambda}$ that I can
obtain from measurements on the degrees of freedom that I can measure
consistent with there being no measurement, I could do, even in principle, on the other
degrees of freedom that could distinguish the particles?''  
This is the best one can hope for under the assumption that we are
limited as to which degrees of freedom we can access.

If the photons are indistinguishable in the hidden degrees of freedom
then of the $D$ possible states that each photon could be in, all $N$
of the photons must be in the same state.  The hidden part of the
state could be written $\ket{\phi}^{\otimes N}$ where $\ket{\phi}$ is a
valid single-particle state in the hidden degrees of freedom.  Such a
state naturally carries the trivial irrep of $S_N$ since if all the
particles are in the same state, any permutation of them will leave the
state unchanged.  From the discussion in the previous section it
follows that the state of the visible degrees of freedom must also
carry the trivial irrep in the visible degrees of freedom.  

If a series of measurements is done that determines all the
$B^\lambda_{q^d_\lambda,q'^d_\lambda}$ and it is found that the only
non-zero elements are those with $\lambda=1$, i.e. those carrying the
trivial irrep of $S_N$ in the visible degree of freedom then we can
say that the measurements are consistent with the $N$ photons being
indistinguishable.  

This does not necessarily mean that there is no measurement that could
be done on the hidden degrees of freedom that would distinguish the
photons.  It only means that if there were no such measurement we
would obtain the same results on the visible degrees of freedom.  The
conclusions of the previous section is that this is the \emph{most}
that we can say about the distinguishability of the photons based on
our measurements.

By the same token, when we measure one of the
$B^\lambda_{q^d_\lambda,q'^d_\lambda}$ with $\lambda \neq 1$ so that
there is a component of the visible degree of freedom that carries a
non-trivial irrep of $S_N$ then we know with certainty that there
is some measurement on the hidden degrees of freedom that could
distinguish the particles.  This has to be the case since we can infer
from the measurement that the hidden degree of freedom also has a
component carrying a non-trivial irrep of $S_N$ which immediately
implies that the photons cannot all be in the same single-particle
state.

Moreover, from the magnitude of the non-trivial $S_N$ component in the
visible degree of freedom we obtain a lower bound on the degree to which
the particles might be distinguishable in the hidden degrees of
freedom.  This is really quite remarkable - our knowledge that photons
are bosons allows us to infer the presence of information in degrees
of freedom \emph{that we can't measure}.
 

\section{Experimental measurements}
With the theory of hidden distinguishability and the accessible density matrix
fully developed we will, in this section, present our reconstructions
of $\rho_{\textrm{acc}}$ for two and three photon polarizations systems.  The
measurements for two photons were completed by Krister Shalm and me in
the summer of 2004.  The three photon data was taken by Krister Shalm
in spring 2007, following the procedure developed in this chapter.  I was only
involved in the data analysis.  

The measurement system used in both experiments was the same.
Conceptually it involved applying to the state an SU(2) operation with
a quarter waveplate and half-waveplate followed by a polarizing
beamsplitter oriented so as to split horizontal and vertical photons
into separate spatial modes and a number resolving detector on each of
these modes.  

The number resolving detector was, in reality, a series of
non-polarizing beamsplitters with a single-photon counting APDs on the
output ports of each beamsplitter.  Figure
\ref{fig:measurement_apparatus} shows a conceptual tomographic apparatus
and the specific beamsplitter networks used for the two and
three-photon characterization.  
\begin{figure}
  \centerline{
   \mbox{\includegraphics[width=\columnwidth]{./Figures/c2f1.eps}}}
  \caption{Measurement apparatuses for measuring the elements of the
    accessible density matrix.  Figure (a) shows an ideal apparatus
    that will work for any number of photons.  Figures (b) and (c) show the
    apparatus actually used for characterizing two-photon and three-photon
    experiments. } 
  \label{fig:measurement_apparatus}
\end{figure}

It may not be immediately apparent that such a system is capable of
measuring all the density matrix elements $B^j_{q^2_j,q'^d_j}$.
However a simple recursive argument shows that this is indeed the
case.  

With $N$ photons in the state there are $N+1$ different
ways that these can split between the $H$ and $V$ ports of the
polarizing beamsplitter so that $N_H$ photons go to the $H$ port and
$N_V$ photons go to the $V$ port.  Clearly $N=N_H+N_V$, and the
angular momentum projection quantum number $m=N_H/2-N_V/2$.  When the
photons leaving each port are counted, the measurement implemented will
be a convex sum of projectors onto all states having that number of
horizontal and vertical photons.
For example, ${\mathbf P}_N=\ket{HH\cdots H}\bra{HH\cdots H}$,
${\mathbf P}_{N-1}=\left(\ket{VH\cdots H}\bra{VH\cdots
    H}+\ket{HV\cdots H}\bra{HV\cdots H}+\ldots\right)$ and so on.
Equivalently, this sum could run over all the states carrying
different $j$ irreps of SU(2) and different irrep of $S_N$ but having
the same value of $m$. 

We begin by looking at the projector ${\mathbf P}_N$.  It has support
only on the $j=N/2$ irrep and is a `pure' projector onto the state $j=N/2$,
$m=N/2$. By changing the angles of the waveplates one can `orbit' this
measurement in the $j=N/2$ space thereby obtaining all the density
matrix element $B^{(N)}_{q^2_{(N)},q'^2_{(N)}}$ of the symmetric
states.    
                              
It follows that the $j=N/2$ part of the accessible density matrix
can be completely characterized by only measuring ${\mathbf P}_N$
rotated under various SU(2) transformations.

${\mathbf P}_{N-1}$ is a convex sum of projectors onto all
states with $m=N/2-1$.  Since the $j=N/2$, $m=N/2-1$ projection has
already been measured, it can
be subtracted off ${\mathbf P}_{N-1}$, leaving a projector with support
only in the $j=N/2-1$ subspace.  By changing the waveplate angles
one can use this modified operator to completely characterize the
$j=N/2-1$ space. One
can then subtract the $j=N/2$ and $j=N/2-1$ terms from the $m=N/2-2$
operator, and so on.  In this way all the terms in the
accessible density matrix can be measured.  The number of operators
obtained in this way will be exactly $\sum_j (2j+1)^2$, or, by
equation \ref{qubitdimensions}, the total number of operators that
\emph{can} be measured on the $N$ photon polarizations.  

In practise the method we followed was to calculate from equation
\ref{qubitdimensions} the number of linearly independent density
matrix elements.  A set of waveplate angles were then chosen and the
projection operators for the measurement apparatus determined as a
function of the waveplate settings.  These projection operators were
treated as vectors in a vector space, and the dimensionality of that
space was calculated by taking the rank of a matrix with the
vectorized projectors as columns.  When the dimensionality of the
space spanned by the vectors matched the number obtained from equation
\ref{qubitdimensions} we knew that we could invert the density matrix
uniquely from the measurements.  The maximum-likelihood techniques
discussed in Chapter 2 were applied to obtain the density matrix from
measurement of these projectors.  

\section{The two-photon experimental polarization density matrix}
We began our investigations into state estimation of indistinguishable
particle states with the two-photon polarization state.  Although at
the time of the experiment we had not developed a full understanding
of permutation symmetries of visible and hidden degrees of freedom,
the two-photon case is simple enough that it can be understood without
needing this formalism.  In group theory language this simplicity
comes about because the permutation symmetries of two particles can be
completely understood in terms of the two one-dimensional irreps of
$S_N$, the trivial irrep and the alternating irrep.  Consequently
there are no values of angular momentum with multiplicity and the
elements of the accessible density matrix all represent coherences or
populations of states with no averaging over multiple spaces with the
same value of $j$.  The two-photon case also has the appealing
property that the effects of distinguishing information are all
concentrated in a single accessible density matrix element, namely the
singlet state projection.

The two-photon case is an important one because
of the many proposals for using two-photon single-spatial mode states
as  a qutrit for quantum
information applications \cite{Bogdanov2004_1, Lang2004,Lanyon2008}.  At the time this experiment was
done, Kulik's group had already developed a quantum state tomography
technique\cite{Bogdanov2004_1,Bogdanov2004_2} for estimating these
states, but, as we have already mentioned, it was flawed in that it
implicitly assumed indistinguishable particles from the outset rather than trying
to obtain all available polarization information and determining
distinguishability from that.

The Schur-Weyl duality applied to two polarizations decomposes the Hilbert space
into a $j=1$ space carrying the symmetric permutation irrep and a
$j=0$ space carrying the alternating permutation irrep.  In order for
the whole state to be symmetric under permutations (which it must be
for bosons), the only available states for the hidden degrees of
freedom are those carrying the same permutation symmetry as the
visible degree of freedom.

A general pure two-photon state can therefore be written as
\begin{equation}
\sum_i
c_{S_i}\ket{{\phi^{j=1}}_i}_{\text{vis}}\ket{{\chi^S}_i}_{\text{hid}}+\sum_k
c_{A_k}\ket{{\phi^{j=0}}_k}_{\text{vis}}\ket{{\chi^A}_k}_{\text{hid}}.
\end{equation}
Here $\ket{{\chi^S}_i}_{\text{hid}}$ represents states on the hidden degrees
of freedom carrying the trivial symmetry irrep and
$\ket{{\chi^A}_k}_{\text{hid}}$ represents states on the hidden degrees of
freedom carrying the alternating irrep.  Similarly  $\ket{{\phi^{j=1}}_i}_\text{vis}$
represents visible states carrying the trivial irrep and
$\ket{{\phi^{j=0}}_i}_\text{vis}$ visible states carrying the
alternating irrep.  The
important point here is that in order to be bosonic the state must be
a sum over terms carrying the same irrep of $S_2$ in the hidden and
visible spaces. 
 
When we trace over the hidden degrees of freedom we are left with 
\begin{align}
\rho_{\text{acc}}={\sum_{m_1,m_2=-1}}^1 c_{m_1} c^*_{m_2}
\ket{\phi^{j=1}_{m_1}}\bra{\phi^{j=1}_{m_2}}+|c_0|^2 \ket{\phi^{j=0}_{m_0}}\bra{\phi^{j=0}_{m_0}}.
\end{align}

This accessible density matrix takes the form 
\begin{equation}
\label{eq:two_photon_dm}
\rho_\text{acc}=\left(\begin{array}
[c]{cc}%
\left(
\begin{array}
[c]{ccc}%
\rho_{HH,HH} & \rho_{HH,\psi^+} & \rho_{HH,VV}\\
\rho_{\psi^+,HH} & \rho_{\psi^+,\psi^+} & \rho_{\psi^+,VV}\\
\rho_{VV,HH} & \rho_{VV,\psi^+} & \rho_{VV,VV}\\
\end{array}
\right) & 0 \\

0 & \left( \rho_{\psi^{-},\psi^{-}}\right)
\end{array}
\right),
\end{equation}

where
$\ket{\psi^+}=\ket{j=1,m=0}=\frac{1}{\sqrt{2}}\left(\ket{HV}+\ket{VH}\right)$
and $\ket{\psi^-}=\ket{j=0,m=0}=\frac{1}{\sqrt{2}}\left(\ket{HV}-\ket{VH}\right)$

In the paper \cite{Adamson2008} we made this argument in a different
way by directly applying the properties of the permutation operators
in $S_2$ without using the Schur-Weyl duality (of which we were
ignorant until later).  The same result as above can be obtained by
writing a completely general two-photon state as
\begin{equation}
 \ket{\psi}=\sum_i c_i
\ket{\phi_i}_\text{acc}\ket{\chi_i}_\text{hid}, \label{thestate}
\end{equation}
where $\ket{\phi_i}_\text{acc},\ket{\chi_i}_\text{hid}$ are
eigenstates of exchange operators ${\bf X}_\text{acc}$ and ${\bf
X}_\text{hid}$ for the visible and hidden degrees of freedom,
respectively, with the same eigenvalue $\pm 1$.  The requirement
that the whole state be bosonic so that ${\bf
X}_\text{acc}\otimes{\bf X}_\text{hid}\ket{\psi} = \ket{\psi}$
guarantees that each term can be written with the visible and
hidden parts of the state either both symmetric or both
anti-symmetric. A completely general state of two photons is a
mixture of states such as $\ket{\psi}$, described by a density
matrix $\rho=\sum_j w_j \ket{\psi_j}\bra{\psi_j}$.

The accessible density matrix in equation \ref{eq:two_photon_dm} can
be thought of as simply a rotated version of the ordinary two-qubit
density matrix, but with `missing' coherences between states of
different $j$.  The absence of these elements can be understood as
being directly due to \emph{inaccessible information} about particle
ordering that is present for qubits, but absent in a system where the
photons are experimentally indistinguishable.  The information
contained in these coherences corresponds very directly to
distinguishing information that could in principle be used to tell the
two photons apart.  For example, consider the missing coherence in (\ref{eq:two_photon_dm})
between the singlet state $\ket{\psi^-}=1/\sqrt{2}\left(\ket{HV}-\ket{VH}\right)$ and triplet state
$\ket{\psi^+}=1/\left(\sqrt{2}\ket{HV}+\ket{VH}\right)$.  Note
that $\ket{HV}=1/\sqrt{2}\left(\ket{\psi^+}+\ket{\psi^-}\right)$ and
$\ket{VH}=1/\sqrt{2}\left(\ket{\psi^+}-\ket{\psi^-}\right)$.  The
ability to distinguish $\ket{HV}$ from $\ket{VH}$, the property
we usually think as distinguishability, amounts to knowing the
relative phase between $\ket{\psi^+}$ and $\ket{\psi^-}$.  In a
complete polarization density matrix for two distinguishable particles
this phase would be contained in the coherence between $\ket{\psi^+}$
and $\ket{\psi^-}$.  Its absence is an expression of the inability of
the photons to be distinguished from the information contained in the
accessible density matrix.

The inaccessibility of the other two missing coherences in
(\ref{eq:two_photon_dm}) can be explained in the same way.  If we knew
the value of the coherences between $\ket{HH}$ and $\ket{\psi^-}$ and
between $\ket{VV}$ and $\ket{\psi^-}$ then, taken together with the
populations of $\ket{HH}$ and $\ket{VV}$ this would tell us the phase
between $\ket{\phi^+}=\frac{1}{\sqrt{2}}\left(\ket{RL}+{LR}\right)$ and
$\ket{\psi^-}=\frac{1}{\sqrt{2}}\left(\ket{RL}-\ket{LR}\right)$ and
between $\ket{\phi^-}=\frac{1}{\sqrt{2}}\left(\ket{DA}+{AD}\right)$ and
$\ket{\psi^-}=\frac{1}{\sqrt{2}}\left(\ket{DA}-\ket{AD}\right)$.  This
is just the information need to tell $\ket{DA}$ from $\ket{AD}$ and
$\ket{RL}$ from $\ket{LR}$, i.e. the distinguishing information between the two photons in the
circular and diagonal bases.  

This interpretation of the `missing' information in the
accessible density matrix can also be made for more than two photons,
although it becomes more complex.  For example, the information needed
to distinguish $\ket{HHV}$, $\ket{VHV}$ and $\ket{VHH}$ is clearly
contained in the coherences between the three $m=1/2$ states in the
accessible density matrix, although since there are three such
coherences there is no longer a single number that one can point to as
uniquely containing that distinguishing information.  


\subsection{Measuring the two-photon experimental
  density matrix}
We measured two-photon accessible polarization density matrices for
two-photon states coming from spontaneous parametric downconversion of
a 405-nm, 50 fs pulse in $\beta$-BBO cut for type-II phasematching in
the collapsed cone geometry\cite{Take2001}.  This system
allowed for thorough experimentation with our new characterization
technique for a few reasons.  First, the two SPDC photons as they emerged
from the crystal were quite indistinguishable and distinguishability
between them could be added by delaying one with respect to the other
by inserting birefringent crystals.  This allowed us to explore
the full range of distinguishability from indistinguishable to fully
distinguishable.  Second, the relatively high detection rates for two
photons made it easy to acquire data within a time shorter than the time
it took the alignment of the system to drift.  Detection rates for
photons were of order $100$ per second and a tomographically complete
dataset could be collected in a few minutes.  This compares to
rates of around $0.1/s$ for the three-photon data shown in the next section.

The waveplate angles used to generate different projective
measurements for two photons are listed in Table
\ref{tab:hidden_differences_wp_settings}.  
\begin{table}[t]
\begin{tabular}
[c]{ccc|ccc|ccc}%
 $h$&$q$&${\bf P}$&$h$&$q$&${\bf P}$&$h$&$q$&${\bf P}$\\
\hline\hline
$0^o$&$0^o$&${\bf P}_{HH}$&$22.5^o$&$0^o$&${\bf P}_{HH}$&$45^o$&$0^o$&${\bf P}_{HH}$\\
$22.5^o$&$45^o$&${\bf P}_{HV}$&$11.25^o$&$0^o$&${\bf P}_{HH}$&$0^o$&$22.5^o$&${\bf P}_{HV}$\\
$45^o$&$22.5^o$&${\bf P}_{HH}$&$22.5^o$&$0^o$&${\bf P}_{HV}$&$22.5^o$&$22.5^o$&${\bf P}_{HH}$\\
$0^o$&$0^o$&${\bf P}_{HV}$
\end{tabular}
\centering \caption{The projective measurements used to determine the
  two-photon accessible density matrix.  The detectors can detect either a
coincidence between two photons in the H mode thereby implementing
the projector ${\bf P}_{HH} \equiv \ket{H H}\bra{H H}$ or
a coincidence between the H and V modes thereby implementing ${\bf
P}_{HV} \equiv \ket{H V}\bra{H V}+\ket{V H}\bra{V
H}$. A quarter- and half-waveplate at angles $q$ and $h$
respectively, placed before the detection apparatus, effectively
rotate the detection operators to $U^{\otimes 2}{\bf
P}\left(U^\dagger\right)^{\otimes 2}$ where ${\bf P}$ is either
${\bf P}_{HH}$ or ${\bf P}_{HV}$ and $U\equiv \exp
\left[i\pi\left({\bf \sigma}_z\cos{2 h}-{\bf \sigma}_x\sin{2
h}\right)\right]\exp \left[\frac{i\pi}{2}\left({\bf
\sigma}_z\cos{2 q}-{\bf \sigma}_x\sin{2 q}\right)\right]$, where
${\bf \sigma}_x$, ${\bf \sigma}_y$ and ${\bf \sigma}_z$ are the
Pauli matrices.} \label{tab:hidden_differences_wp_settings}
\end{table}
Note that ten measurements are required to measure the
ten different elements of the accessible density matrix.

The complete experimental setup including both the preparation of the
state and the tomographic measurement apparatus is shown in Figure
\ref{fig:two_photon_experimental_setup}.
\begin{figure}
  \centerline{
    \mbox{\includegraphics[width=\columnwidth]{Figures/figure1}}
  }
  \caption{Experimental implementation of state preparation and tomography protocols
  showing polarizing beamsplitters (PBS), beamsplitters (BS), non-linear $\beta$-Barium
  Borate (BBO) crystals, a second harmonic generation crystal (SHG), quarter
  waveplates (QWP), half-waveplates (HWP), polarization maintaining fiber (PMF), single-mode fiber
  (SMF) and single photon counting modules (SPCM).
  A spontaneous parametric downconversion (SPDC) crystal produces pairs of H and V photons. The
  separation between the H and V photons is controlled with
  movable quartz wedges and very long delays can be introduced by inserting a thick piece of BBO
  into the beam.  Single-mode fiber
  and a 10nm interference filter make the photons essentially indistinguishable
  in the spatio-temporal modes.  Tomography is performed with a set of waveplates
  and a polarizing beamsplitter.
  This system can implement all the measurements in Table \ref{tab:hidden_differences_wp_settings}.}
  \label{fig:two_photon_experimental_setup}
  \end{figure}

50 fs pulses centered at 810 nm  were generated in a Ti:Sapph
oscillator with
a 90 MHz repetition rate.  The average power of the pulse train was
400 mW.  These pulses
were upconverted at a 0.5 mm BBO crystal to create 405-nm
centered pulses which were sent to a second 0.5 mm BBO crystal to be
downconverted.  The downconversion was phase-matched for collapsed
cone `beamlike' emission so that the horizontal and vertical photons
left the crystal in individual beams separated by a $6^\circ$ angle.
These two photons were then recombined at a polarizing beamsplitter so
as to travel in a single spatial mode.  The path lengths of the H and
V photons were balanced to within much less than a coherence length
(30 $\mu$m as determined by the filter bandwidth) so that the two
photons were in wavepackets centered at the same point in time.

To ensure minimal hidden distinguishability, filters, were used to select a
narrow range of frequencies and a single spatial mode.  The photons
were passed through a 10 nm full-width at half-maximum spectral filter
which significantly reduced the spectral distinguishability.

A single-mode optical fiber selected a single spatial mode for the
photons which guaranteed that any remaining hidden distinguishing information
would be in the temporal degree of freedom.  This temporal
distinguishability could be manipulated by changing the group delay
between the H and the V modes.  When this delay was well in excess of
the coherence time of the photons they were distinguishable, since a
measurement of arrival time would then give complete information about
`which' photon had which polarization.  When the delay was non-zero
but less than the coherence time, a measurement of the arrival time
could only give partial information about the polarization.

\begin{figure}
  \centerline{
   \mbox{\includegraphics[width=\columnwidth]{Figures/figure2.eps}}}
  \caption{The density matrices measured with quantum state tomography.  The imaginary parts of (a) through (f),
  which are not shown, had all elements less than 0.05.
  White elements are inaccessible to measurement.
  (a) indistinguishable H and V photons
  (b) distinguishable H and V photons
  (c) partially distinguishable H and V photons
  (d) indistinguishable photons transformed to the 2-NOON state
  (e) distinguishable photons with the same transformation
  applied
  (f) the same state as it would be characterized by the
  technique of \cite{Bogdanov2004_2}}
   \label{fig:hidden_differences_dms}
\end{figure}


\begin{table}
\footnotesize
\begin{tabular}{l}
(a)$\left(
\begin{array}{cccc}
0.020 & 0.025+0.022i & 0.001+0.016i  & \\
0.025-0.022i &  0.949  & -0.016-0.056i & \\
-0.001-0.016i & -0.016+0.056i & 0.014  \\
& & & 0.017
\end{array}
\right)$  
\\
(b)$\left(
\begin{array}{cccc}
0.047 & -0.066+0.041i & 0.003+0.023i & \\
-0.066-0.0410i  &  0.620 & 0.002+0.004i & \\
0.003-0.023i  & 0.002-0.004i & 0.021 & \\
& & & 0.312
\end{array} 
\right)$
\\
(c)$\left(
\begin{array}{cccc}
0.021  & -0.038+0.002i & 0.002-0.034i  & \\
-0.038-0.002i &  0.499 & 0.009+0.000i & \\
0.002+0.034i & 0.009+0.000i & 0.024 & \\
& & & 0.456
\end{array}  
\right)$ 
\\
(d)$\left(
\begin{array}{cccc}
0.264  & 0.006+0.021i & 0.225-0.027i &\\
0.006-0.021i  & 0.029  & -0.017-0.000i & \\
0.225+0.027i & -0.017+0.000i & 0.256 & \\
& & & 0.450
\end{array}
\right)$
\\
(e)
$\left(
\begin{array}{cccc}
0.459 & 0.042+0.022i  & 0.418+0.038i &\\
0.042-0.022i & 0.057  & -0.008+0.002i & \\
0.419-0.022i & -0.008-0.002i & 0.484 & \\
& & & 0.0000
\end{array}
\right)$ 
\\
(f)$\left(
\begin{array}{ccc}
0.264 & 0.0248+0.032i & -0.233+0.028i \\
0.025-0.032i &  0.489 & -0.011+0.030i \\
-0.233-0.028i & -0.011-0.030i & 0.247 
\end{array}
\right)$
\end{tabular}
\normalsize
  \caption{The accessible density matrices of various two-photon
    polarization states measured with quantum state tomography.  
  (a) indistinguishable H and V photons
  (b) distinguishable H and V photons
  (c) partially distinguishable H and V photons
  (d) indistinguishable photons transformed to the 2-NOON state
  (e) distinguishable photons with the same transformation applied
  (f) the same state as it would be characterized by the technique of
  \cite{Bogdanov2004_2}}
   \label{tab:hidden_differences_dms}
\end{table}

Measurements were taken by setting the waveplates in figure
\ref{fig:hidden_differences_krister_apparatus} to the ten values in table
\ref{tab:hidden_differences_wp_settings}.  This resulted in ten
linearly independent projections from which the accessible density
matrix element could be extracted.  Maximum-likelihood fitting was
achieved by fitting to a lower-triangular matrix $T$ of the form
\begin{equation}
T=\left(
\begin{array}{cccc}
t_1 & 0 & 0 & 0\\
t_2+i t_3 & t_4 & 0 & 0\\
t_5+i t_6 & t_7+i t_8 & t_9\\
0 & 0 & 0 & t_{10}\\
\end{array}
\right),
\end{equation} 
so that $\rho_\text{acc}=T^\dagger T/\text{Tr}\left[T^\dagger
  T\right]$.  A least-squares likelihood function was minimized
over all values of $t_i$ using Mathematica's
FindMinimum function.

Six measured density matrices are shown in Figure
\ref{fig:hidden_differences_dms} and Table
\ref{tab:hidden_differences_dms}.  The state was initially prepared to
contain one $H$ photon and one $V$ photon with zero path length
difference.  The zero-path length difference point was found by the
observation of two-photon interference in the diagonal basis.  When
tomography was carried out on this state, the density matrix in Figure
\ref{fig:hidden_differences_dms}(a) was obtained.  $95\%$ of the population was found
to be in $\ket{\psi^+}$ and less than 2\% in the $\ket{\psi^-}$ state.  On
the other hand when a delay much larger than the photon coherence
length was introduced between the two polarizations, the density
matrix in Figure \ref{fig:hidden_differences_dms}(b) was obtained.  The
population was split roughly evenly between the
$\ket{\psi^+}$ and $\ket{\psi^-}$ states.  

The difference between the two density matrices becomes further apparent when
the state is rotated using a quarter waveplate with its axis at
$45^\circ$ to the horizontal.  When the two polarizations are
indistinguishable this results in the state
$\frac{1}{2}\left({a^\dagger}_H {a^\dagger}_H+{a^\dagger}_V
{a^\dagger}_V\right)$, or equivalently the 2-N00N state\cite{Boto2000} or the
$\ket{\phi^+}$ Bell state.  Our measured density matrix for this state
is shown in figure \ref{fig:hidden_differences_dms}(d).  This state
has zero probability of
producing anti-correlated photons in the $H$/$V$ basis.  On the other
hand when the distinguishable photons of figure
\ref{fig:hidden_differences_dms}(b) are sent through the same quarter
waveplate, the resulting state still has a $j=0, m=0$ component
consisting of half of the population as shown in figure
\ref{fig:hidden_differences_dms}(e).  This ensures that
anti-correlated counts will be observed in every basis half of the
time.  The two states really are different, and any good state
estimation scheme must measure that difference.  

In fact, if we try to use the state characterization scheme
proposed in \cite{Bogdanov2004_2} to estimate the state in
\ref{fig:hidden_differences_dms}(e) we obtain completely
different results.  We can use our reconstructed density matrix to predict the
outcome of measurements used in reference \cite{Bogdanov2004_2}.
The result is the $3 \times 3$ density matrix shown in Table
\ref{tab:hidden_differences_dms}(f).  Their reconstruction procedure
lumps the two $m=0$ populations together into the $j=1,m=0$
state and incorrectly measures the coherence between the $m=1$
and $m=-1$ states.  This density matrix makes completely incorrect
predictions about correlation measurements in all bases except
the H/V basis.

Figure \ref{fig:hidden_differences_dms}(c) shows what happens when the
distinguishing delay is non-zero. but less than a coherence
length.  While both the
$j=0, m=0$ and $j=1, m=0$ populations are non-zero, the $j=1, m=0$
term is larger.  In fact, the $j=0, m=0$ term can be thought of as a
direct measurement of the distinguishability to the extent that
the distinguishability affects polarization measurements.  Two
photons both polarized along $H$ can be distinguishable in some
other degree of freedom, but this distinguishability will have no
effect on polarization measurements.  On other hand making two
photons distinguishable when one is $H$ and the other is $V$ will
always affect polarization measurements because the distinguishing
degree of freedom is correlated to polarization.  It is the degree
of correlation between polarization and all the other degrees of
freedom of the photons that is measured in the $j=0, m=0$ density
matrix element.

In summary, ours is the only two-photon polarization state
tomography scheme that is able to properly account for
distinguishability so as to correctly predict the outcomes of all
measurements.  It deals equally well with indistinguishable,
distinguishable and partially distinguishable photon states.

\subsection{Measuring the three-photon accessible density matrix}
The three-photon case proceeds in a similar way to the two-photon
case, but three-photon polarization states offer a much richer state
space than two photon states.  Part of this richness manifests itself
in the more complicated characterization needed.  Since $S_3$ has
multi-dimensional irreps, the density matrix contains not one, but two
$j=1/2$ sectors that characterize the distinguishability.  There are
four different numbers in this submatrix.  The total population of the
two $j=1/2$ submatrices plays a role similar to the $j=0, m=0$
component for two photons, while the distribution of populations and
the coherences within the $j=1/2$
submatrices also contains additional information about the
polarization.  

Even if we forget about the complexities of $S_3$ and consider only
the case of indistinguishable photons we obtain a richer set of
states.  For two polarizations the N00N is equivalent to the Dicke
state with $m=0$, but for three photons these are distinct states.  In
fact, a range of coherent, squeezed, and Dicke states are available
with three photons.  Some of these states have features that
make them attractive for quantum metrology applications\cite{Durkin2007}.

We undertook state estimation for a wide range of
three-photon polarization states\cite{Shalm2008} using the
techniques developed in this chapter.  To create the states we employed
the apparatus pictured in figure
\ref{fig:hidden_differences_krister_apparatus}.  A two-photon
polarization state was created in SPDC using the same setup as in the
previous section, but then a third photon was added to it from an
attenuated laser pulse (marked LO for local oscillator in the
figure).  This addition of the third photon is
achieved by a process known as mode-mashing which involves
simply bringing photons together on a beamsplitter and
post-selecting on events where all the photons leave in one port.  The
process is non-deterministic, but when it succeeds there is a clear
experimental signal of its success.  It can be shown that if, after
mashing, the photons are spatio-temporally indistinguishable, then any
three-photon polarization state can be created simply by polarizing
three photons and mode-mashing them together.  The polarization of the
state was controlled unitarily with waveplates and also non-unitarily
with a variable partial polarizer (VPP).  The ability to perform
non-unitary transformations meant that the relative angle between the
polarizations could be changed.

In the real experiment true single-photon sources were not used, resulting
in some probability that the post-selection would incorrectly signal
success if, for example, three photons from the laser pulse exited the
same output port or if two SPDC pairs did.  By adjusting the relative
power of SPDC and the attenuated laser pulse these `background' counts
could be minimized, resulting in the post-selection correctly selecting
for the desired state around $60\%$ of the time.  More details of the
experiment can be found in \cite{Shalm2008}.   Here we will
discuss the characterization of the states.

\begin{figure}
  \centerline{
   \mbox{\includegraphics[width=\columnwidth]{Figures/krister_apparatus.eps}}}
  \caption{Apparatus for measuring the three-photon accessible density matrix}
  \label{fig:hidden_differences_krister_apparatus}
\end{figure}

To characterize the three-photon polarization state, the state was
directed to a polarizing beamsplitter to separate $H$ and $V$ photons.
The $V$ output was further sent to two cascaded beamsplitters.
Assuming perfect detector, the
presence of three photons at the $V$ port would trigger a coincidence
between detectors B, C and D $7.3\%$ of the time.  Similarly, when one photon left
in the $H$ port and two in the $V$ port, it was signaled by coincident
firing of two of detectors B, C and D and detector A.  If the detector
were perfectly efficient this would happen $45\%$ of the time.
Relative detector efficiencies could also be calibrated out so that
the rate of three-fold coincidences between different detectors could
be used to measure the average number of times three photons were
present at the $V$ port of the PBS, thereby projecting onto $m=-3/2$, and the number of times that two photons were
present at the $V$ port and one at the $H$ port, thereby
projecting onto $m=-1/2$.

A quarter and half-waveplate placed before the polarizing beamsplitter
allowed these two measurements to be rotated into an informationally
complete set of projection operators capable of spanning all the
elements of the accessible density matrix.  The outcomes of these
projective measurements were then fed into the maximum-likelihood convex
problem solver SeDumi\cite{Sturm1999}.  The accessible density matrix was constrained to
take the block-diagonal form of equation
\ref{formofrhoacc} with the two $j=1/2$ subspaces
constrained to be equal.  

Data from a typical experimental tomography run is
shown in Table \ref{tab:noondata}.  
\begin{table}
\label{tab:noondata}
\begin{tabular}{|cc|cc||cc|cc|}
\hline
\multicolumn{2}{|c|}{Settings} & \multicolumn{2}{c||}{Counts} &
\multicolumn{2}{c|}{Settings} & \multicolumn{2}{c|}{Counts}\\
QWP & HWP & m=-1/2 & m=-3/2 & QWP & HWP & m=-1/2 & m=-3/2\\ 
\hline\hline 
$0^\circ$ & $0^\circ$ & 26815 & 40665 & $0^\circ$ & $30^\circ$ & 40867 & 24970 \\
$15^\circ$ & $0^\circ$ & 17861 & 28894 & $15^\circ$ & $30^\circ$ & 49587 & 14625 \\
$30^\circ$ & $0^\circ$ & 14598 & 34958 & $30^\circ$ & $30^\circ$ & 29173 & 32104 \\
$45^\circ$ & $0^\circ$ & 27852 & 37455 & $45^\circ$ & $30^\circ$ & 27367 & 26040 \\
$0^\circ$ & $15^\circ$ & 34970 & 36741 & $0^\circ$ & $45^\circ$ & 51006 & 28894 \\
$15^\circ$ & $15^\circ$ & 30571 & 32818 & $15^\circ$ & $45^\circ$ & 41763 & 14982\\
$30^\circ$ & $15^\circ$ & 14737 & 34958 & $30^\circ$ & $45^\circ$ & 66353 & 6778\\
$45^\circ$ & $15^\circ$ & 25858 & 32461 & $45^\circ$ & $45^\circ$ & 15664 & 36741\\
\hline
\end{tabular}
\caption{Collected counts from a 3-photon tomography in an experiment
 attempting to make the NOON state $\ket{1}{\sqrt{2}}\left(\ket{3_H,0_V}+i\ket{0_H,3_V}\right)$.}
\end{table}

After the detector
efficiencies were taken into account, this data was fit to the
accessible density matrix below
\begin{changemargin}{-1.5cm}{-1.5cm}
\tiny
\begin{equation*}
\left(
\begin{array}{cccccccc}
0.2448 & 0.0057+0.0008i & -0.0323i & 0.0325-0.1437i &  &  &  & \\ 
0.0057-0.0008i & 0.1577 & 0.0819-0.0885i & -0.0653-0.0286i & & &
&\\
0.0323i & 0.081908+0.0885i & 0.1763 & 0.0609-0.0600i & & & &\\
0.0325+0.1437i & -0.0653+0.0286i & 0.0609+0.0600i & 0.2639 & & &
&\\
& & & & 0.0494 & 0.0220+0.0297i & &\\
& & & & 0.0220-0.0297i & 0.0293 & &\\
& & & & & &  0.0494 & 0.0220+0.0297i\\
& & & & & & 0.0220-0.0297i & 0.0293
\end{array}
\right).
\label{eq:noondm}
\end{equation*}
\normalsize
\end{changemargin}

The non-ideality of the state preparation is apparent from the
population in the two $j=1/2$ subspaces which amounted to $16\%$ of
the total population of the state.  Unlike in the two-photon
experiment where the population of the $j=0$ could be
essentially eliminated by
temporally overlapping the photons, in the three-photon experiment the particles
could not be made indistinguishable.  This is because the two
photons generated via SPDC are fundamentally distinguishable from the
photon taken from the laser due to temporal broadening from the SHG
and SPDC processes and because of residual spectral entanglement due
to phase-matching in the SPDC.  All of these effects can be corrected
with stronger spectral filtering, and experiments to test the
dependence of the $j=1/2$ populations on spectral filtering are
ongoing.  

Within the $j=3/2$ space, the state looks like a N00N state, with
strong coherences between the $\ket{m=3/2}$ and $\ket{m=-3/2}$ terms.
The extra diagonal population can be largely accounted for as being
due to the unwanted background coincidence sources.  When these were
subtracted from the measured density matrix the result is quite a bit closer
to a N00N state\cite{Shalm2008}.

\section{Summary and conclusions}
We have developed a complete theory of quantum state tomography for
indistinguishable particles.  This theory elegantly handles the
reduction in the size of the Hilbert space when particles are
fundamentally indistinguishable and provides a lower-bound measure of just how
distinguishable they are.  It is the most complete state
characterization possible for experimentally indistinguishable
particles.  The theory was applied to two and three-photon
states, and their accessible density matrices were presented. 

It is hoped that the techniques outlined in this
chapter will become the standard characterization method for this class
of state, which has applications in quantum metrology,
lithography and orienteering, just as ordinary quantum state
tomography\cite{James2001} has become standard for distinguishable particles.
